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Luke_A_Somers comments on The Problem of "Win-More" - Less Wrong Discussion
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This is part of a general issue in binary win/loss games.
As you approach the end of a game with a lead in points, the value of scoring more points diminishes. If you don't adjust your strategy to compensate, you end up taking risks that increase your expected points, but decrease your win probability. Win probability plotting is popular for American Football obsessives.
It's often very frustrating to be a sports fan, as it's obvious what's going wrong with the decision theory in the heads of coaches and players.
I wonder if other sports could use a model similar to the Triwizard Tournament. The outcome of the first two events sets the handicap for the third event, which is the only one that really counts. Be ready for some bad ideas.
Adapting this to American Football, it could be that your score after the first three quarters is converted into an advantage in downs or yards in some fashion for the last quarter.
Baseball - each run in the first 8 innings counts as an extra out in the last inning.
In Everyone Else Football... umm... I'm having a hard time coming up with something not boring but less insane than "each goal you score in the first three quarters is an extra ball that goes on the field in the last quarter and can only be scored into your opponents' goal".
This isn't very much “a sport”, but it came to mind that Team Fortress 2's Payload Race game mode works exactly like this on multi-stage maps. The handicap (starting farther ahead/behind) is very small and usually overwhelmed by team organization and the outcomes of combat, though.
I would watch this variant of Everyone Else Football.
In a failed attempt to make the game "interesting", the NFL Pro Bowl would do the opposite, and help the team that was behind. They would get the ball again after they had scored if they were still behind. Naturally, that makes it easier to catch up.